Fast implementation of a maximum likelihood algorithm for the estimation of target motion parameters
A system and method for implementing a maximum likelihood estimator for making a joint estimation of range, range rate, and acceleration of a target utilizing a pulse doppler radar. The MLE of target motion parameters are determined by keystone processing a baseband signal, and generating a first estimate of the motion parameters based on the processed signal. The first estimate is utilized to set up sampling intervals for the performance of a coarse search. Then a fine search is performed using Newton's method to determine the MLE.
1. A method for determining a maximum likelihood estimate (MLE) of motion parameters of a target from a return signal from a pulse doppler radar, comprising:
keystone processing a baseband signal;
generating a first estimate of the motion parameters;
utilizing the first estimate to perform a coarse search of an ambiguity function Ψ to determine a second estimate of the motion parameters corresponding to global maxima of samples of Ψ at predetermined sampling intervals; and
performing a fine search by applying Newton's method to Ψ to find the MLE, utilizing the second estimate of the motion parameters as an initial condition.
2. The method of claim 1 , further comprising
transmitting a linear frequency modulated radar pulse train toward the target; and
receiving and preprocessing the return signal reflected from the target to obtain the baseband signal.
3. The method of claim 2 , wherein the ambiguity function Ψ has three dimensions, such that the motion parameters comprise a range r, a range rate ν, and an acceleration a of the target, wherein the range r is proportional to a round trip delay τ between the transmitting and the receiving.
4. The method of claim 3 , wherein keystone processing the baseband signal comprises translating the doppler frequency of the baseband signal into a keystone-processed signal based on the equation:
f
d
′
=
(
f
RF
+
k
Δ
f
f
RF
+
K
Δ
f
)
f
d
where f d is a doppler frequency, f RF is a carrier frequency of a transmitted waveform, Δf is a frequency step, k is a range frequency index, and f′ d is a translated doppler frequency.
5. The method of claim 4 , wherein the first estimate of the motion parameters comprises a first range estimate and a first range rate estimate, and wherein the first estimate is extracted from the keystone-processed signal.
6. The method of claim 5 , wherein the ambiguity function Ψ comprises a correlation function between a real phase of the return signal and a model phase.
7. The method of claim 6 , wherein the coarse search comprises:
setting up a coarse sampling grid for sampling the ambiguity function Ψ in parameter space; and
sampling Ψ over the coarse sampling grid.
8. The method of claim 7 , wherein the predetermined sampling intervals are a grid spacing of the coarse sampling grid, and wherein the grid spacing is proportional to a mainlobe width of the ambiguity function Ψ in each of its three dimensions.
9. The method of claim 8 , wherein the grid spacing in parameter space is given by the equation:
(
τ
,
v
,
α
)
=
ζ
(
±
1.51
π
c
2
1
(
2
K
+
1
)
Δ
f
,
±
1.51
π
λ
2
1
(
2
N
+
1
)
I
P
P
,
±
0.000249
λ
(
81
(
2
N
+
1
)
I
P
P
)
2
)
,
wherein c is the speed of light in a vacuum; λ is a wavelength of the return signal; K is an integer such that (2K+1)Δf is a bandwidth of the transmitted waveform; N is an integer and IPP is an inter-pulse period such that (2N+1)IPP is a dwell duration; and ζ is a constant for scaling the grid spacing.
10. The method of claim 9 , wherein ζ is approximately in the range of 0.5≦ζ≦0.75.
11. The method of claim 9 , wherein the fine search comprises repeating a recurrence of
(
τ
k
+
1
v
k
+
1
α
k
+
1
)
=
(
τ
k
v
k
α
k
)
-
[
Ψ
(
2
)
(
τ
k
,
v
k
,
α
k
)
]
-
1
Ψ
(
1
)
(
τ
k
,
v
k
,
α
k
)
starting at the coarse search selected grid point until a Cramer-Rao bound (CRB) is reached for each of the target motion parameters, where Ψ (1) is a gradient of Ψ, and Ψ (2) is a Hessian of Ψ.
12. The method of claim 1 , wherein the recurrence is repeated while Δτ=τ k+1 −τ k ≦τ CRB is true; or while Δν=ν k+1 −ν k ≦ν CRB is true; or while Δα=α k+1 −α k ≦α CRB is true.
13. The method of claim 12 , further comprising performing fine motion compensation, wherein
W k,n (τ,ν,α)= Ŷ k,n e j2πf RF τ n (τ,ν,α) ×e j2πkΔfτ n (τ,ν,α) ,
where W k,n is a fine motion compensated radar return, and Ŷ k,n is a frequency-domain representation of the baseband signal.